MinT - Best Known (109−83, 109, s)-Nets in Base 27

Best Known (109−83, 109, s)-Nets in Base 27

(109−83, 109, 114)-Net over F₂₇ — Constructive and digital

Digital (26, 109, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 109, 114)-net over F₂₇, using
- net from sequence [i] based on digital (23, 113)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 23 and N(F) ≥ 114, using
    - an explicitly constructive algebraic function field [i]

(109−83, 109, 208)-Net over F₂₇ — Digital

Digital (26, 109, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 109, 208)-net over F₂₇, using
- net from sequence [i] based on digital (24, 207)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 24 and N(F) ≥ 208, using
    - an algebraic function field by Beelen and Pellikaan [i]

(109−83, 109, 3637)-Net in Base 27 — Upper bound on s

There is no (26, 109, 3638)-net in base 27, because

1 times m-reduction [i] would yield (26, 108, 3638)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38905 229823 028668 401673 275871 041552 118479 277812 068506 740107 361696 761087 709366 703837 533913 480363 629265 593078 169569 864995 109086 636817 992668 043900 777026 731949 > 27¹⁰⁸ [i]